EXTRAPOLATION FILTERS

The explicit 3-D migration techniques consist of two stages. In the first stage, the filters for two-dimensional extrapolation are built and stored in a table. In the second stage the coefficients of the extrapolation filters are used along with the McClellan filters in the Chebyshev filter structure to extrapolate the wavefield in three dimensions.

We design our extrapolation filters for the migration using the following equation,  

$$D(k)= \exp{\left \{ i\frac{\Delta{z}}{\Delta{x}}\left[\sqrt{... ...{\frac{\omega \Delta x}{v}}^2 \right ) -k^2 }\right] \right \}}$$ (1)

and windowing techniques Nautiyal et al. (1993); Parks and Burrus (1987). We used a gaussian taper in the space domain. Although the filters are not optimized for a given dip and velocity model, they are designed to be stable. The McClellan extrapolation requires a look-up table of filters parametrized in terms of $\omega dx /v$.The filters are parametrized for $\omega dx /v$ , so for a given v(z) velocity model, the filters are designed for the range $\omega_{min} dx /v_{max}$ to $\omega_{max} dx / v_{min}$.